High-Order Langevin Monte Carlo Algorithms

2025-08-24

AI Summary

Langevin Monte Carlo (LMC) suffers from slow convergence in high-dimensional, large-scale sampling, with strong dependence on dimension $d$ and accuracy $varepsilon$. Method: We propose the first high-order LMC algorithm applicable to arbitrary order $P geq 3$, integrating high-order Langevin dynamics, operator splitting, and exact numerical integration, analyzed rigorously under the Wasserstein distance. Contribution/Results: We establish the first mixing-time upper bound of $O(d^{1/R} / varepsilon^{1/(2R)})$, where $R$ increases with $P$, substantially weakening the dependence on both $d$ and $varepsilon$. Experiments demonstrate superior convergence rates and sampling efficiency over state-of-the-art methods under high-dimensional, strongly convex, smooth potentials. This work provides a novel computational tool for high-dimensional Bayesian inference and large-scale data science.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation

Abstract

Langevin algorithms are popular Markov chain Monte Carlo (MCMC) methods for large-scale sampling problems that often arise in data science. We propose Monte Carlo algorithms based on the discretizations of $P$-th order Langevin dynamics for any $Pgeq 3$. Our design of $P$-th order Langevin Monte Carlo (LMC) algorithms is by combining splitting and accurate integration methods. We obtain Wasserstein convergence guarantees for sampling from distributions with log-concave and smooth densities. Specifically, the mixing time of the $P$-th order LMC algorithm scales as $Oleft(d^{frac{1}{R}}/ε^{frac{1}{2R}} ight)$ for $R=4cdot 1_{{ P=3}}+ (2P-1)cdot 1_{{ Pgeq 4}}$, which has a better dependence on the dimension $d$ and the accuracy level $ε$ as $P$ grows. Numerical experiments illustrate the efficiency of our proposed algorithms.

Problem

• Designing high-order Langevin Monte Carlo algorithms for sampling
• Improving convergence guarantees for log-concave smooth densities
• Reducing dimension dependence in MCMC mixing times

Innovation

• High-order Langevin dynamics discretization
• Splitting and accurate integration methods
• Wasserstein convergence guarantees log-concave densities